This book provides a comprehensive account of the statistical theory of exponential families of stochastic processes. The book reviews the progress in the field made over the last ten years or so by the authors, two of the leading experts in the field, and several other researchers. The theory is applied to a broad spectrum of examples. The statistical concepts are explained carefully so that probabilists with only a basic background in statistics can use the book to get into statistical inference for stochastic processes. Exercises are included to make the book useful for an advanced graduate course.